Operator ${\ell }_p\to {\ell }_q$ norms of Gaussian matrices

Abstract

We confirm the conjecture posed by Guédon, Hinrichs, Litvak, and Prochno in 2017 that $\mathbb{E}\Vert (a_{ij}{g}_{ij}{)}_{i\le m,j\le n}:{\ell }_p^n\to {\ell }_q^m\Vert$ is comparable, up to constants depending only on $p$ and $q$, to \[ \max_i \|(a_{ij})_j\|_{p^*} +\max_j \|(a_{ij})_i\|_{q} +\mathbb{E} \max_{i,j} |a_{ij}g_{ij}| \] provided that $1\le p\le 2\le q\le \infty$. This was known before only in the case $p=1$ or $q=\infty$, and in the spectral case $p=2=q$. We also reprove the conjecture in the case $p=2=q$ without using spectral theory (which was employed in the previously known proof).

42 pages; final version; typos corrected, references updated, Remark 15 fixed, some minor changes introduced